Question

What are the four rules of binary addition?

Hint:

When performing binary addition, remember to keep track of the carry-over, especially when adding ones. The carry is added to the next higher bit in the sequence.

Answer 1

Binary addition is a fundamental operation in binary arithmetic, where we add two binary numbers following a set of specific rules. These rules are similar to decimal addition but are simpler because each digit is either 0 or 1. The Four Rules of Binary Addition: When adding two binary digits (bits), the following rules apply: \[ \begin{array}{|c|c|c|} \hline Binary Addition & Sum & Carry
\hline 0 + 0 & 0 & 0
0 + 1 & 1 & 0
1 + 0 & 1 & 0
1 + 1 & 0 & 1 \, (\text{carry})
\hline \end{array} \] Explanation of the Rules: 1. 0 + 0 = 0 : Adding two zeros results in a sum of 0, with no carry. 2. 0 + 1 = 1 or 1 + 0 = 1 : Adding a zero and a one results in a sum of 1, with no carry. 3. 1 + 1 = 0 with a carry of 1: Adding two ones results in a sum of 0, but there is a carry of 1 that is added to the next higher bit in the addition. 4. Carry Over : When the sum of two bits results in 2 (which is 10 in binary), a carry of 1 is generated, and the sum for that digit becomes 0.
Example:
Consider adding the two binary numbers \( 1011_2 \) and \( 1101_2 \): \[ \begin{array}{c} \phantom{+}1011_2
+ 1101_2
\hline \phantom{+}11000_2
\end{array} \] Let's break down the addition step-by-step: \[ \begin{array}{|c|c|c|} \hline \text{Bit Position} & \text{Adding} & \text{Result}
\hline \text{1st bit} & 1 + 1 & 0 \, (\text{carry 1})
\text{2nd bit} & 0 + 0 + 1 \, (\text{carry}) & 1
\text{3rd bit} & 1 + 1 & 0 \, (\text{carry 1})
\text{4th bit} & 1 + 1 + 1 \, (\text{carry}) & 1
\hline \end{array} \] The final sum is \( 11000_2 \), with a carry of 1 at the 5th bit position. Summary:
The four rules of binary addition are: 1. \( 0 + 0 = 0 \) (no carry) 2. \( 0 + 1 = 1 \) or \( 1 + 0 = 1 \) (no carry) 3. \( 1 + 1 = 0 \) with a carry of 1 4. The carry is added to the next higher bit position.